CN107076778B - High-speed self-adaptive multi-loop mode imaging atomic force microscope - Google Patents

Abstract

A method of imaging a sample using a high speed dynamic mode atomic force microscope may include scanning a tip of a cantilever probe over a surface of a sample; adjusting the vibration amplitude of the tip via a first signal generated in a first feedback controller so as to be kept constant at a set point value (A)_set) (ii) a Measuring an average tap deflection of the tip; adjusting the mean tap offset via a second signal generated in a second feedback controller; adjustments to the measured mean tap offset during the adjustment are tracked and measured. The method also includes generating an image topography of the sample based on the first signal, the second signal, and the measured adjustment to the mean tap offset of the cantilever probe.

Description

High-speed adaptive multi-loop mode imaging atomic force microscopy

technical field

The present disclosure generally relates to operating atomic force microscopy (AFM) at increased speed in one or more dynamic modes (DMs) while preserving the inherent properties of ultra-high pixel quality and reducing probe-sample interaction forces for each mode Systems and methods for dynamic modes including, but not limited to, percussion mode (TM) and non-contact mode (NCM) and peak force mode (PFM) and contact mode (CM). More specifically, the present disclosure relates generally to systems and methods for accounting for changes in the mean shift (or shift in a contact pattern) of a probe in quantifying sample topography, using and incorporating: (i) Inner-outer feedback loops to adjust the average cantilever offset around the minimum level required to maintain stable probe-sample interactions in the corresponding imaging modalities; (ii) an online iterative feedforward controller; (iii) On-line optimization of vibration amplitude ratio; and (iv) for probe vibration generators (which minimize the set-point of vibration amplitude) and for rms probe vibration amplitude feedback controllers feedback controller.

Background technique

AFM is a high-resolution scanning probe microscope with resolution in the nanometer range. in an atomic force microscope. In atomic force microscopy, a microcantilever with a sharp tip (probe) at its end can be used to scan the surface of a sample. As the tip begins to approach the sample surface, according to Hooke's law, the force between the tip and the sample may cause a deflection of the cantilever. Typically, the deflection of the cantilever is measured to obtain the topography of the sample.

AFM can be run in a variety of imaging modes including contact mode (CM, also known as static mode) as well as a variety of dynamic modes including, but not limited to, TM, PFM, and NCM. In dynamic mode imaging, the cantilever is driven to oscillate vertically with a fixed oscillation amplitude. The amplitude of the oscillations may decrease due to the interaction forces acting on the cantilever as the tip approaches the sample surface. In conventional DM imaging, a microprocessor, digital signal processor or field programmable gate array (FGPA) based system and an underline control algorithm are often used to control the height of the cantilever above the sample so that the A fixed oscillation amplitude (as in TM and NCM imaging) or a fixed peak repulsion amplitude (as in PFM imaging) is maintained while scanning across the sample surface. The RMS displacement of the cantilever in the vertical direction is used to generate images of the sample topography in the DM microscope, assuming that the cantilever oscillation amplitude is well maintained at the ideal set-point value during the scan.

Compared to CM imaging techniques, DM imaging mode imaging generally provides better image quality and lower sample distortion due to reduced capillary forces, friction and shear forces, and contact pressure. However, DM imaging speed tends to slow down considerably, as increasing the imaging speed can result in a loss of probe-sample interaction and/or attenuation of cantilever knock vibrations (especially when the sample size is larger).

Like most measurement devices, AFMs often require a trade-off between quality and acquisition speed. That said, some currently available AFMs can scan surfaces at sub-Angstrom resolution. These scanners are only capable of scanning relatively small sample areas, however, only at relatively low scan rates. For example, conventional commercial TM imaging AFM typically requires a total scan time of typically up to ten minutes to cover an area of several microns at high resolution (eg, 512 x 512 pixels) and low probe-sample interaction forces. The imaging speed of PFM is generally comparable to that of TM, while the imaging speed of NCM is generally slower than that of TM. This is mainly because TM and PFM operate in the repulsive force layer or the middle layer between the repulsive force and the attractive force region, but the NCM operates purely in the attractive force region and the probe hovers further above the sample surface, and the probe-sample The interaction force is more sensitive to the probe-sample distance in the gravitational region. Because probe-sample interaction forces are more sensitive to probe-sample spacing in the region of attraction, non-contact mode imaging tends to be slower than tapping mode imaging. The practical limit to AFM scan speed is the maximum speed at which the sample can be scanned while keeping the probe-sample interaction force low enough to not damage the tip and/or sample or cause non-negligible damage to the tip and/or sample the result of.

Therefore, there is a need for high-speed DM imaging and/or CM imaging techniques with controllable interaction forces and suitable for imaging large size samples.

SUMMARY OF THE INVENTION

A method of imaging a sample using a high-speed dynamic mode atomic force microscope is disclosed. The method may include: scanning the tip of the cantilever probe over the surface of the sample; adjusting the vibration amplitude of the tip via a first signal generated in a first feedback controller so as to remain constant at a set point value (A _set ) ; measure the average tap excursion of the tip; adjust the average tap excursion via a second signal generated in a second feedback controller; track and measure the average tap excursion measured during adjustment adjust. The method also includes generating an image topography of the sample based on the first signal, the second signal, and the measured adjustment to the average tap offset of the cantilever probe. In an embodiment, the high-speed dynamic mode atomic force microscope may include one or more of the following: a tapping mode, a non-contact mode, and a peak force tapping mode.

In some embodiments, adjusting the average tap offset may include: using the ratio of A _set to free amplitude to determine a desired average offset, and adjusting the measured average tap offset to the desired Average offset. In an embodiment, the desired average excursion is determined such that the ratio of A _set to free amplitude is between about 10% and 30%.

In some embodiments, the second feedback controller may include an inner and outer feedback loop structure. An outer feedback loop can adjust the average tap offset, and an inner loop nested within the outer loop can perform tracking and measurement adjustments to the measured average tap offset during adjustment. In an embodiment, the outer feedback loop is a proportional-integral-derivative (PID) type controller with PID parameters _Kp , _KI and _KD . In at least one embodiment, the PID-type controller may employ the following algorithms:

d _TM-set (j+1)=k _I d _TM-set (j)+k _P e _TM (j)+k _D [e _TM (j-1)-e _TM (j)]

where e _TM (j) = d _{TM - d -} d _TM (j), where j = 2...N-1

In an embodiment, the PID parameters have the following values: K _P =1, K _I =1 and K _D =ρ, where ρ is the sample pointwise gradient factor. In an embodiment, p<1.

Alternatively and/or additionally, the PID-type controller employs the following algorithms:

d _{set, 0} = d _{set, org} ,

其中t∈[0，T_scan]，in

where t∈[0, _Tscan ],

In some embodiments, the method may further include optimizing _Aset online based on the measured average excursion and the real-time relationship between _Aset and the vibration amplitude ratio of the free amplitude. The method may also include predicting, via a third feedback controller, a next-line sample topography and a next-line tracking error for tracking the average tap offset adjustment. In some embodiments, the method may further include using the next row of sample topographies and the next row to track erroneous predictions for regions with sample surfaces that provide features that provide sudden dynamic changes, including cliffs and edges reduce tracking errors. In at least one embodiment, using the next row of sample topography and the prediction of the next row of tracking errors in order to reduce tracking errors comprises obtaining the desired trajectory for the next row using the following formula:

h _{ffd, k+1} (j)=h _k (j)+α[d _{TM, k} (j)-d _TM-d ], j=1, . . . N _l .

In one embodiment, the value of a may be adjusted based on the estimated height of the feature.

Optionally and/or additionally, the feedforward controller may further include a zero-phase low-pass filter configured to filter noise from being fed back to the feedforward controller.

In some embodiments, the third feedback controller may be a feedforward controller, including a data-driven iterative learning controller. The feedforward controller implements the following algorithms to obtain control inputs:

U _ff,0 (jω)=0,

e _k (jω)=H _{ffd, k+1} (jω)-Z _k (jω)

In some embodiments, obtaining the desired trajectory for the next line may also include performing repetitive scans of the first line until convergence is reached, and using the convergence as an initial input for an iteration of the next scan line.

In another aspect of the present disclosure, a method of imaging a sample using a high-speed dynamic mode atomic force microscope is disclosed. The method may include scanning the tip of the cantilever probe over the surface of the sample; adjusting the vibration amplitude of the tip via a first signal generated in a first feedback controller so as to remain constant at a set point value ( _Aset ). The method may further include measuring an average tap offset of the tip; adjusting the average tap offset via a second signal generated in a second feedback controller; and tracking and measuring the measured change during adjustment. Adjustment of the average tap offset; predicting the next-line sample topography and next-line tracking error for tracking the average tap offset adjustment via a third feedback controller; using all the predicted next-line sample topography and the next-line tracking error; and generating the sample based on the first signal, the second signal, and the measured adjustment to the average tap offset of the cantilever probe image shape.

In certain embodiments, the method may also include on-line iterative control applied to the z piezoelectric actuator to maintain a stable tap. In an embodiment, applying the on-line iterative control includes adjusting A _set by on-line point-by-point adjustment.

In certain embodiments, adjusting the average tap offset may include: using the ratio of A _set to free amplitude to determine a desired average offset; and adjusting the measured average tap offset to the desired Average offset. In an embodiment, the desired average offset is determined such that the ratio of A _set to free amplitude is in the range of about 10% to 30%.

In some embodiments, the second feedback controller may include an inner and outer feedback loop structure. An outer feedback loop adjusts the average tap offset, and an inner loop nested within the outer loop may perform tracking and measurement of adjustments to the measured average tap offset during adjustment. In an embodiment, the outer feedback loop is a proportional-integral-derivative (PID) type controller with PID parameters _Kp , _KI and _KD . In at least one embodiment, the PID-type controller employs the following algorithm:

d _TM-set (j+1)=k _I d _TM-set (j)+k _P e _TM (j)+k _D [e _TM (j-1)-e _TM (j)]

where e _TM (j) = d _{TM - d -} d _TM (j), where j = 2...N-1

In an embodiment, the PID parameters have the following values: K _P =1, K _I =1 and K _D =ρ, where ρ is the sample pointwise gradient factor. In an embodiment, p<1.

Alternatively and/or additionally, the PID-type controller may employ the following algorithms:

d _{set, 0} = d _{set, org} ,

其中t∈[0，T_scan]，in

where t∈[0, _Tscan ],

In some embodiments, the method further comprises using the next row of sample topography and the next row to track erroneous predictions in areas with sample surfaces that provide features that provide sudden dynamic changes, including cliffs and edges Reduce tracking errors. In at least one embodiment, using the next-row sample topography and the prediction of the next-row tracking error in order to reduce the tracking error may include obtaining the desired trajectory for the next-row using the following formula:

h _{ffd, k+1} (j)=h _k (j)+α[d _{TM, k} (j)-d _TM-d ], j=1, . . . N _l .

In an embodiment, the value of a may be adjusted based on the estimated height of the feature.

Optionally and/or additionally, the feedforward controller may further include a zero-phase low-pass filter configured to filter noise from being fed back to the feedforward controller.

In some embodiments, the third feedback controller may be a feedforward controller, including a data-driven iterative learning controller. The feedforward controller may implement the following algorithms to obtain control inputs:

U _ff,0 (jω)=0,

e _k (jω)=H _{ffd, k+1} (jω)-Z _k (jω)

In some embodiments, acquiring the desired trajectory for the next line may further include performing repetitive scans on the first line until convergence is achieved; and using the convergence as an initial input for an iteration of the next scan line.

Description of drawings

The present disclosure will be better understood and objects other than those set forth herein will become apparent when reference is made to the following detailed description herein. Such description refers to the accompanying drawings, in which

Figure 1A shows a block diagram of an exemplary prior art DM imaging AFM microscope.

Figure IB shows a block diagram of an exemplary prior art contact mode imaging AFM microscope.

1C is a graph showing an exemplary relationship of probe-sample interaction force with respect to probe-sample distance in a TM imaging AFM microscope according to an embodiment.

2 is a schematic diagram of an exemplary sample surface topography including two sample points and the offset of the TM offset at the two sample points, according to an embodiment.

3A is a diagram of an exemplary adaptive multi-loop mode imaging AFM control block diagram of the present disclosure in accordance with a first embodiment.

3B is a diagram of an exemplary adaptive multi-loop mode imaging AFM control block diagram of the present disclosure, according to a second embodiment.

3C is a diagram of an exemplary adaptive multi-loop mode imaging AFM control block diagram that may be incorporated into a contact mode AFM according to a third embodiment of the present disclosure.

3D is a diagram of an exemplary adaptive multi-loop mode imaging AFM control block diagram of the present disclosure, which may be incorporated into percussion mode AFM, according to a fourth embodiment.

4 is a graph illustrating a cliff-induced spike in an average offset according to an embodiment and the decrement of the cliff-induced spike using the iterative feedforward control of the present disclosure.

5 is a flowchart illustrating a method for adaptive multi-loop mode imaging for quantifying sample topography, according to an embodiment.

6 is a graph showing an exemplary relationship between the average TM offset amplitude and the ratio of the RMS vibration amplitude to the free vibration amplitude of the cantilever.

7 is a schematic diagram of an exemplary atomic force microscope system, according to an embodiment.

Detailed ways

It will be readily understood that the components of the embodiments generally described herein and illustrated in the accompanying drawings may be arranged and designed in many different configurations. Thus, as indicated in the figures, the following detailed description of the various embodiments is not intended to limit the scope of the present disclosure, but is merely representative of the various embodiments. Although the drawings illustrate various aspects of the embodiments, the drawings are not necessarily drawn to scale unless otherwise indicated.

The present disclosure may be embodied in other specific forms without departing from the spirit or essential characteristics of the present disclosure. The described embodiments are considered to be illustrative in all respects. Accordingly, the scope of the present disclosure is indicated by the appended claims. All changes that come within the meaning and range of equivalency of the claims are included within the scope of the claims.

Reference throughout this specification to features, advantages, or similar language does not imply that all features and advantages that may be realized by virtue of the present disclosure should or are present in any single embodiment of the present disclosure. Rather, language referring to features and advantages may be considered to mean that a particular feature, advantage or characteristic described in the relevant embodiment is included in at least one embodiment of the present disclosure. Thus, discussions of the features, advantages, and similar language throughout this specification may, but are not necessarily, considered to be the same embodiment.

Furthermore, the features, advantages and characteristics described in this disclosure may be combined in any suitable manner in one or more embodiments. From the description herein, one skilled in the relevant art will appreciate that the present disclosure may be practiced without one or more of the specific features or advantages of a specific embodiment. In other instances, other features and advantages may be realized in certain embodiments that may not be mentioned in all embodiments of the present disclosure.

Reference throughout this specification to "one embodiment," "an embodiment," or similar language means that a particular feature, structure, or characteristic described in connection with the associated embodiment is included in at least one embodiment of the present disclosure. Thus, the words "in one embodiment," "in an embodiment," and similar language throughout this specification may, but do not necessarily, all refer to the same embodiment.

As used in this document, the singular forms "a," "an," and "the" include plural referents unless the context clearly dictates otherwise. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art. As used in this document, the word "including" means "including but not limited to".

The present disclosure provides, by way of example only, applications of embodiments for increasing the speed of TM mode imaging. It should be understood by those skilled in the art that embodiments may also be used to increase the speed of other kinds of AFM, such as PFM or NCM, for example, without departing from the principles of the present disclosure.

Referring now to FIG. 1A, an exemplary prior art DM imaging AFM block diagram is provided. As mentioned above, the cantilever probe in DM imaging is activated to continuously vibrate. Vibration can cause the probe tip to interact with the sample surface 105 . Vibration demodulator 308 may be used to measure relative amplitudes or magnitudes (eg, RMS tap amplitude in TM imaging, peak force amplitude (ie, repulsive force amplitude) in PFM imaging, and RMS oscillation amplitude in NCM imaging). In the case of TM or NCM imaging, the lock-in amplifier 109 can be used to measure the vibration root mean square (RMS) amplitude at any specified time. In PFM imaging, a differentiator can be used to measure the maximum repulsive amplitude of probe vibration at various time periods. The amplitude of the vibration can vary according to the sample topography, because the interaction of the forces acting on the probe as it approaches the surface (van der Waals forces, interactions between dipole moments, electrostatic forces, etc.) decreases as the tip approaches the sample. A feedback controller 102 (eg, an RMS z-feedback controller) can be utilized to keep the amplitude constant at the desired setpoint value _Aset throughout the imaging process. The feedback controller may use the piezoelectric actuator 103 to keep the amplitude constant at _Aset . Displacement in piezoelectric actuator 103 can be measured to quantify sample topography. In dynamic mode imaging and PFM imaging, probe wear and/or sample damage caused by probe sliding over the sample surface (as in CM mode imaging) can be significantly avoided, resulting in higher sample resolution .

As shown in Figure IB, the probe-sample interaction force, which in turn controls the amplitude, is non-linear and a function of probe-sample distance. Therefore, changes in probe-sample distance at higher imaging speeds may result in attenuation of vibrations (tapping) or loss of probe-sample contact.

Typically, in DM imaging, multiple time periods of vibration need to be acquired to measure amplitude (eg, tap amplitude in TM imaging, peak force amplitude in PFM imaging, or oscillation amplitude in NCM imaging). This can induce a time delay in the feedback control loop, and the measured amplitude can be different (lag) from the actual instantaneous tap amplitude of the cantilever probe. The adverse effect of the time delay on the feedback controller 102 becomes negligible only at low scan speeds.

As an example, in the prior art, the TM imaging (it should be noted that TM mode imaging is used only as an example) module has been selected by selecting a large free vibration amplitude (A _free ) and a small tapping amplitude set point (A _set ) and try to increase the imaging speed. However, this results in large probe-sample interaction forces, as state-of-the-art modules typically ignore changes in cantilever offset and offset, which results in image distortion at high imaging speeds. The formula that determines the relationship between the probe-sample interaction force and cantilever deflection is:

F _ts (t)=-k _c d _tot (t)=-k _c [d _TM (t)+(A _def cos(ω ₀ t+φ))-A _free cos(ω ₀ t)] ——( 1)

where k _c is the elastic constant of the cantilever, d _tot (t) is the total deflection of the cantilever, d _TM (t) is the average cantilever deflection per unit time period (TM-offset), and A _def (t) is the instantaneous knock The strike amplitude, φ and _Afree are the phase shift of the cantilever response to excitation and the cantilever free amplitude, respectively.

Referring now to Figure 2, a typical sample topography quantification is shown. Examples of samples may include poly-tert-butyl acrylate (PtBA), polypropylene-low density polyethylene (PS-LDPE), styrene (SBS), polypropylene film (celgard), and biological samples. The sample size can be selected from approximately 1 square micrometer (sq. μm) to 10,000 square micrometers. Samples can be prepared for AFM imaging using techniques known in the art, which are not disclosed here for simplicity. Considering the TM offsets (205 and 206), the height difference between any two points (203 and 204) on the sample surface (202) can be expressed as:

h _1-0 =[z(x ₁ ,y ₁ )–z(x ₀ ,y ₀ )]+ε[d _TM (x ₁ ,y ₁ )–d _TM (x ₀ ,y ₀ )]--- -----(2)

in:

z(x,y)=z piezoelectric displacement at point (x,y);

d _TM (x, y) = average offset at point (x, y); and

ε = Contact constant depending on the probe-sample interaction mechanism. When the probe-sample interaction is dominated by a distant attractive force (eg, A _def /A _free ∈ (0.5, 0.8)), ε = -1, when the repulsive probe-sample interaction force occurs ε = 1, and -1<<ε<0, when the percussion amplitude is close to the free vibration amplitude, that is, when A _def ≈A _free .

Thus, the above formula (2) means that the topography of the sample in the entire imaging area can be acquired with respect to a fixed reference point (eg, point 0—referred to as the first imaging sample point for convenience). Without loss of generality, in some embodiments, the height and offset reference point 0 may be set to z(x ₀ , y ₀ ) = 0 and d _TM (x ₀ , y ₀ ) = d _{TM - d} , where d _TM-d is the average deviation of the probe vibration amplitude corresponding to the set point value, and thus the sample surface topography can be quantified as:

h(x,y)=z(x,y)+ε[d _TM (x,y)-d _TM-d ]=z(x,y)+εΔd _TM (x,y)------ --(3)

Therefore, at low imaging speeds, the cantilever probe can accurately track the sample topography in it under the control of the vibration amplitude (ie, A _def closely surrounds the setpoint value, and the average offset Δd _TM (x,y) varies enough small and d _TM (x,y)≈d _TM-d ). Thus, h(x,y)≈z(x,y), and the sample topography can be accurately quantified as piezoelectric displacements measured at each sample point.

However, maintaining the strict condition of A _def ≈ A _free is difficult as the imaging speed increases. Note that even with a small scan speed increase (small change in vibration amplitude), the change in the instantaneous amplitude A _def cannot be ignored, so the change in the mean offset may still be noticeable. Such mean shift changes are not typically calculated in conventional DM imaging, and cause image distortions and limit imaging speed.

Therefore, as described in this disclosure, imaging speed can be increased by considering the mean shift in sample topography quantification.

Figure IB shows a contact mode AFM block diagram and may differ slightly from the dynamic mode of imaging. First, in contact mode, the probe tip is "dragged" across the surface of the sample, and the deflection of the cantilever is used directly, or more generally, the deflection of the cantilever is maintained (i.e., the probe-sample interaction force) The desired feedback signal closely around the set point value can measure the profile of the surface. Because measurements of static signals can be prone to noise and drift, a low-stiffness cantilever is used to improve the offset signal. Close to the surface of the sample, the attraction can be quite strong, causing the tip to "bite" to the surface. Thus, contact mode AFM is almost always performed at a depth where the force is generally repulsive, ie a firm "contact" with the solid surface beneath any adsorption layer. Furthermore, in contact mode imaging, the vibrational modes of the cantilever may not be excited, and sample deformation can be assumed to be negligible. Therefore, unlike DM imaging, d(x,y) in contact mode imaging may represent shift rather than average shift, and the contact constant ε may not be required. Considering these differences between the contact mode and DM mode of the imaging, the height difference between any two points (203 and 204) on the sample surface (202) can be expressed as:

h _1-0 =h _1-0 =[z(x ₁ ,y ₁ )–z(x ₀ ,y ₀ )]+[d(x ₁ ,y ₁ )–d(x ₀ ,y ₀ )]- -------(2A)

in,

z(x ₀ , y ₀ ) = the z-axis position of the sample point (x ₀ , y ₀ );

z(x ₁ , y ₁ ) = the z-axis position of the sample point (x ₁ , y ₁ );

d(x ₀ , y ₀ ) = cantilever offset at sample point (x ₀ , y ₀ ); and

d(x ₁ , y ₁ ) = cantilever offset at sample point (x ₁ , y ₁ ).

Thus, by choosing z(x ₀ , y ₀ ) = 0 and d(x ₀ , y ₀ ) = d _set (set-point deflection), the sample surface topography can be quantified as:

h(x,y)=z(x,y)+[d(x,y)-d _set ]-------(3A)

Therefore, imaging speed can be increased by accounting for shift in sample topography quantification, as described in this disclosure.

In a first embodiment, the present disclosure describes an AFM with improved imaging speed that includes an imaging module that, in addition to a feedback loop (as discussed with respect to FIG. 1A ) that adjusts vibration amplitude while maintaining probe vibration, may The average cantilever excursion is adjusted using a control loop of the inner and outer loop configuration to operate. In another embodiment, a data-driven online iterative feedforward controller can be integrated into the inner and outer loop structures to further improve the tracking of sample topography.

In a first approach, the present disclosure may increase imaging speed in adaptive multi-loop imaging (AML imaging) AFM by calculating and adjusting the vibration of the cantilever offset as discussed below.

Referring now to FIG. 3A, a block diagram of an exemplary AML imaging module for enhancing imaging speed in AFM is provided. As shown in Figure 3, the AML imaging module may include: (i) a vibration amplitude feedback control loop (300); (ii) a feedback control (320) in the inner and outer loop structures for adjusting the mean offset; (iii) vibration amplitude online optimization of the ratio (314); (iv) a feedback controller for the probe vibration generator (which minimizes vibration amplitude) (370); and (v) an online iterative feedforward controller (310) for Overcome the time delay of the vibration amplitude feedback loop when tracking sample topography.

The vibration amplitude control loop 320 may use a vibration demodulator (eg, a lock-in amplifier in the case of TM imaging) 304 and the vibration amplitude controller 301 discussed earlier with respect to FIG. 1A to adjust the AML imaging vibration amplitude at a set point value . The vibration demodulator 303 can measure the average vibration amplitude or the peak force amplitude, and the feedback controller 301 can use the z piezoelectric actuator 302 to adjust the amplitude. Techniques and related techniques for implementing such control loops are well known in the art and, therefore, will not be described in further detail herein, except for details that may be helpful or required to understand the operation of the system.

In addition to the vibration amplitude feedback controller, the inner and outer loop structure 320 can also maintain stable rapping by adjusting the average offset by adjusting the average (vertical) position of the cantilever to closely surround the ideal value during each rapping cycle. In particular, the outer loop 350 can adjust the average offset setpoint in real time, and the inner loop 360 can use the controller 306 to track the adjusted average offset setpoint. Controller 306 may be a feedback controller such as a PID-type controller.

The outer loop can use the following PID-type (proportional-integral-derivative) control to adjust the average offset set point d _TM-set ( ):

d _TM-set (j+1)=k _I d _TM-set (j)+k _P e _TM (j)+k _D [e _TM (j-1)-e _TM (j)]

where e _TM (j) = d _{TM - d -} d _TM (j), where j = 2...N-1, - (4)

in:

N = total number of sampling periods for each image;

d _TM-d = expected mean offset;

d _TM (j) = average offset of the current sampling point; and

K _P , K _I and K _D are PID parameters.

The desired mean offset d _TM-d can be judged by the ratio of the selected tap amplitude set point to free amplitude, A _set /A _free . In particular, the relationship between d _TM-d and A _def /A _free can be measured in advance. The relationship of d _TM-d to A _def /A _free may resemble a parabolic curve centered on A _def /A _free ~50%.

Although the present disclosure uses a PID controller, it will be understood by those skilled in the art that different types of controllers may be used without departing from the principles of the present disclosure.

In an embodiment, the PID parameters K _P =1, K _I =1 and K _D =ρ, where ρ may be the sample pointwise gradient factor. In an embodiment, p<1. It should be understood that these PID values are exemplary values and may be adjustable for better performance.

FIG. 6 depicts a plot 600 of an exemplary d _TM-d versus A _def /A _free for the tap mode case, where the tip-sample interaction force is significant when the tap ratio A _def /A _free < 10% Increase, when A _def /A _free is greater than 80%, increasing the scan speed may quickly lead to detachment. According to an embodiment, for the curve of d _TM-d versus A _def /A _free as shown in Figure 5, the desired average offset can be chosen such that the corresponding A _def /A _free is preferably between 10% and 30% (601). Although the present disclosure is based on the graph shown in Figure 6, it will be understood by those skilled in the art that different taps may be preferred for different samples and/or cantilevers without departing from the principles of the present disclosure ratio, depending on the specific measured d _TM-d versus A _def /A _free .

Furthermore, it should be appreciated that there may be other ways in which the outer loop can adjust the average offset setpoint in real time. The foregoing lists only a limited number of examples to which the embodiments herein are not limited.

It should be noted that according to equations (1) and (3), although the average shift can be calculated separately in sample topography quantification to increase the speed of AML imaging AFM, it does not improve the tracking of sample topography during imaging. Therefore, as the imaging speed increases, the probe-sample interaction force may change significantly and lead to disengagement and/or elimination of knocking. In a second aspect, the present disclosure thus discloses on-line iterative feedforward control implemented on the z piezoelectric actuator 302 in order to keep track of the sample topography.

The _set point of the probe RMS vibration amplitude (Aset in Figure 3A, Figure 3B, and Figure 3D) can be adjusted to calculate the uncertainty compared to the RMS vibration amplitude relationship and the change in mean offset (as shown in Figure 6) . The RMS vibration amplitude at steady state and the mean offset d _TM can be measured in real time during imaging and then used to construct the portion of the mean offset versus amplitude curve around the chosen desired mean offset d _{TM - d} . The steady state mean offset _dTM and RMS vibration amplitude can be obtained by averaging the measured data over a period of time greater than the time constant of the mean offset feedback control loop and the time constant of the RMS vibration amplitude loop, respectively. Then, based on the real-time measured d _TM vs. A _def /A _free curve, the amplitude setting optimization controller 314 can be adjusted by plotting the d _TM vs. A _def /A _free curve corresponding to the desired average shift d _TM The value of _-d sets the vibration amplitude setpoint _Aset to update the setpoint for the RMS vibration amplitude _Aset .

As shown in FIG. 3A , an online iterative feedforward controller 308 corresponding to the z piezoelectric actuator 302 may be integrated into a vibration amplitude feedback loop 330 . Specifically, the feedforward control input is obtained by implementing the following high-order model-free differential inversion-based iterative control (HOMDIC) algorithm online:

U _{ff, 0} (jω)=0, ——(5)

e _k (jω)=H _{ffd, k+1} (jω)-Z _k (jω) (8)

in:

λ = preselected constant to ensure convergence of iterations;

参见图3； _Uff+fb,k (·)=total control input (feedback+feedforward) applied to the z piezoactuator, i.e.,

See Figure 3;

Z _k (·) = z piezoelectric displacement measured at scan line k; and

H _ffdk+1 (·)=z piezoelectric predetermined track that needs to be tracked at the k+1th scan line.

和

本质上等于z压电致动器的频率响应的逆变换，并且可以在整个成像过程中逐行反复地更新。迭代方案的中的预先获得固定模式上的这种数据驱动的在线更新的逆变换可以出现更好的鲁棒性和跟踪性能。最终，时间域U_ff+fb,k+1(t)中前馈输出可以经由傅里叶逆变换来获得并在第k+1行的扫描期间被使用。The ratio in the above control rule

and

Essentially equal to the inverse transform of the frequency response of the z-piezo actuator, and can be iteratively updated row by row throughout the imaging process. The inverse transform of this data-driven online update on a pre-obtained fixed pattern in an iterative scheme can yield better robustness and tracking performance. Finally, the feedforward output in the time domain _Uff+fb,k+1 (t) can be obtained via an inverse Fourier transform and used during the scan of row k+1.

In addition, for those skilled in the art, it should be understood that there are other methods in which the online feedforward control loop 310 can control the sample topography tracking, and the foregoing content only lists an exemplary algorithm, and the embodiments of the present invention are not limited to these examples.

The average shift can respond more quickly to changes in sample topography than the tap amplitude. However, due to the compliance of the cantilever and the cantilever clamp (connecting the cantilever to the piezoelectric actuator), there may still be a time delay between the average shift change and the topographic profile change. Such time delays, albeit small, may become important as the scan speed increases, so that the average shifted peaks may not reach their (local) peaks until the probe has passed these sample locations. Even with advanced feedback control, such offset spikes may persist.

Thus, according to an embodiment, the feedforward controller 308 in the AML imaging module may employ a data-driven iterative control algorithm (eg, the algorithms given by Equations 5-8) to significantly improve the tracking of sample topography during high-speed imaging . Additionally and/or alternatively, the feedforward controller 308 may use predictions of sample topography and sample tracking errors for the next row in order to significantly reduce sample regions with sudden and drastic changes (eg, with Tracking errors (ie, cantilever offset changes) near the region of the cliff and/or edge shown). Specifically, by combining a data-driven iterative controller with a prediction-based predetermined trajectory of the next row (as discussed previously), the feedforward controller 308 can significantly reduce cantilever deflection near areas with cliffs and/or edges shift changes. As shown in Fig. 4, large spikes appear near the "cliff" and "edge" of the block-shaped multi-segment sampling area. The amplitude of these spikes can be significantly reduced by using data-driven iterative feedforward control, which can then be further reduced by incorporating tracking error predictions in the feedforward control, as shown in Figure 4.

The predicted sample topography and predicted next row mean offset tracking error may be calculated by the feedforward controller 308 to track the modified predetermined trajectory H _ffd,k (·), as shown in Equation 7. The modified predetermined trajectory (feedforward control data for the trajectory) enables the z-piezo to drive the cantilever to respond earlier (ie, pre-drive) to the topography changes, thus reducing the amplitude of the excursion spikes. For example, at the end of scan line k, the sample topography profile of line k+1 can be predicted using the sample topography profile of scan line k (quantified via Equation 5) (ie, h _k+1 (t)≈h _k (t)). Such an approximation is reasonable because with enough scan lines, the topography varies little from row to row. Similarly, if the same control is applied to the same sample topography, the average offset tracking error on the k+1th scan line can be predicted as the average offset tracking error on the kth line d _TM,k ( ) –d _TM-d (ie, tracking error). Then, the predetermined trajectory H _ffd,k+1 (t) of the next row can be obtained by combining the aforementioned two predictions, as follows:

h _{ffd, k+1} (j)=h _k (j)+α[d _{TM, k} (j)-d _TM-d ], j=1, . . . N _l . ——(9)

in,

N _l = total sample points per image line; and

a=correction factor.

An average offset can be introduced in the iterative algorithm described above (Equation 9) to reduce the amplitude of the interaction force when imaging sample regions with steep and large topographical changes (vertical changes). In some embodiments, the correction factor α may be adjusted based on the estimated height of the sample surface features.

In an embodiment, the iterative scheme defined above may be reused during imaging to scan the first row until convergence is reached (ie, until the z-piezoelectric displacement difference between two consecutive iterations is sufficiently small, eg, close to noise, level (noise level). In some embodiments, 2 to 3 repetitive scans may be performed on the first row to be able to scan the rest of the sample without iteration. In other embodiments, 7 to 8 repetitive scans may be performed on the first row. The converged input can then be used as the initial input for the iteration on the next scanline. Keeping the correction rate (ie, convergence rate) of the iterative input faster than the line-by-line input changes caused by sample topography changes, it may be possible to update the iterative control input only once (ie, the rest of the sample can be imaged without iteration). This use of z-piezodynamics can provide a larger "working" bandwidth (ie, better tracking performance at high speeds) because feedback control tends to reduce open-loop bandwidth.

In some embodiments, to avoid noise feedback into the closed loop via the feedforward channel, the feedforward control input _Uff,k+1 (·) may be passed through a zero-phase low-pass filter

in:

以及

as well as

Since the entire next row feedforward control input is known inherently, the above noncausal zero-phase filter can be implemented online.

In an embodiment, A _def /A _free may be equal to 20%, so that d _TM-d may be close to zero. In other embodiments, A _def /A _free may be equal to 30%, such that d _TM-d may be close to zero. In other embodiments, A _def /A _free may be between 20% and 30%, such that d _TM-d may be close to zero.

It should be noted that while the speed of the AML imaging AFM can be increased by calculating the mean offset and performing feedforward tracking of the sample topography during imaging, the tap amplitude is always kept constant at the predetermined value _Aset . Therefore, imaging speed can be increased and probe-sample interaction forces can be changed, which can lead to unstable tapping and increased noise in the signal. Thus, in a third aspect, the present disclosure discloses an on-line iterative control applied to the z piezoelectric actuator 302 in order to maintain a stable tap.

3B illustrates an exemplary AML imaging module block diagram in accordance with an embodiment of the present disclosure. As shown in Figure 3B, the vibration amplitude feedback control loop 330 shown in Figure 3A can be modified by adding a feedback loop 340 to further reduce probe-sample interaction forces by adaptively adjusting the tap amplitude.

In some embodiments, the vibration amplitude (ie, the tap amplitude in TM imaging, the peak force amplitude in PFM imaging, or the vibration amplitude in NCM imaging) can be tightly maintained to maintain a stable tap and ideal signal-to-noise than the required minimum amplitude. In particular, the outer loop 340 may use the controller 301 to adjust the set point of the vibration amplitude A _set on-line and point by point (ie, A _set may not be a predetermined constant). In some embodiments, the free oscillation amplitude A _free is adjusted point by point, and then the corresponding A _set is determined based on the relationship between A _free and A _set at any given average "average offset" value previously measured. Adjust A _set . The controller 301 may be a feedback controller, such as a PID-type controller.

In some embodiments, controller 301 may be a PID-type controller that adjusts _Aset using the following formula:

A _free (j+1)=k _ia A _free (j)+k _pa e _a (j)+k _da [e _a (j-1)-e _a (j)]

where e _a (j)=A _min -A _def (j), where j=2...N-1,——(11)

in:

N = total number of sampling periods for each image;

A _min = the lower limit of the desired vibration amplitude to maintain the desired signal-to-noise ratio and stable tapping of the cantilever;

A _def (j) = vibration amplitude at the current sampling point; and

Kia, _Kpa and _Kda are PID control parameters.

The desired vibration amplitude set point A _set (j+1) at sampling point (j+1) can be determined based on a pre-measured relationship that will use the sampling point (j) determined by equation (11) using the following formula: The free vibration amplitude A _free (j+1) at +1) is related to the mean offset d _TM-d :

A _set (j+1)=f(A _free (j+1),d _TM-d ) ------------(12)

in:

f(A _free (j+1), d _TM-d ) = a function that defines the relationship of A _free to A _set at the given mean offset value.

The relationship can be predetermined, ie an explicit expression of f(A _free (j+1),d _TM-d ) may not be required, and can be determined experimentally to obtain a numerical relationship.

In some embodiments, the desired average offset may be chosen as a constant as close to zero as practical.

It should be noted that while this disclosure illustrates the incorporation and use of AML imaging modules in various dynamic modes of AFM (eg, tapping mode), AML imaging modules may also be used without departing from the principles of this disclosure. Incorporated in contact mode AFM for increased imaging speed. 3C illustrates a block diagram of an exemplary AML imaging module incorporated into a contact mode AFM, according to an embodiment of the present disclosure.

As discussed earlier in Figure 1B, in contact mode imaging, imaging speed can be increased by accounting for offset in sample topography quantification. Thus, as shown in Figure 3C, in contact mode imaging, the inner and outer feedback loops (325) can adjust the offset set point line by line (as opposed to point by point in DM imaging) to maintain the offset, thereby connecting probe-sample The interaction force is kept near the minimum level required to maintain stable contact during the scan. Specifically, the outer loop 355 can adjust the offset set point in real time, and the inner loop 365 can use the controller 365 to track the adjusted offset set point. Controller 365 may be a feedback controller, such as a PID-type controller.

The outer loop can adjust the offset setpoint using gradient-based minimization of the normal force as described below:

d _{set, 0} = d _{set, org} , ——(13)

其中t∈[0，T_scan]，——(15)in

where t∈[0, T _scan ],——(15)

in:

d _set,0 = offset set point on the first scan line;

d _set,k = offset set point on scan line k;

d _k+1 (t)=minimum of predicted detections at scan line k+1;

D* _min = minimum deflection/force (ie, threshold) required to maintain a stable repulsive end-sample interaction;

T _scan = scan period;

d _set,org = original offset set point selected prior to the imaging process; and

ρ∈[0,1]=gradient factor, which can be adjusted to improve imaging quality.

In contact mode imaging, the data-driven iterative feedforward control loop 310 for the piezoelectric actuator 302 can operate in the same way as the feedforward control loop in the DM imaging module discussed earlier. For example, the desired trajectory H _ffd,k+1 (t) for the next row can be obtained using Equations 5, 6 and 9.

3D illustrates a block diagram of an exemplary AML imaging module for use in a tap-mode AFM, according to embodiments of the present disclosure. The imaging module may function in the same manner as shown with respect to the imaging module of Figure 3A.

In some embodiments, the signals disclosed above may be acquired by a data acquisition system, such as in a MatlabxPC target environment. Other examples may include DSP-based data acquisition and computing systems, FPGA-based data acquisition and computing systems, or any other similar systems known in the art.

In conclusion, the advantages and efficacy of the AML imaging techniques of the present disclosure relative to current imaging techniques, including DM imaging and CM imaging, can be explained and understood using the experimental results shown in Figure 4, which compares the use of current imaging techniques and AML imaging of a calibration sample with multiple segments in a square shape. Figure 4A shows a cross-section of the topography of the sample to be imaged. Accurately tracking the edges of segments can be difficult due to the stepped sample features. For example, as shown in Figure 4B, in the cantilever excursion signal (ie, the change in probe-sample interaction force), large spikes may appear on the upper and lower edges of the sample when using conventional imaging techniques with only feedback. Consequently, image contours measured using conventional imaging techniques (shown in Figure 4C) cannot capture the edges of the sample contour (compare Figure 4A with Figure 4C). However, as shown in Figure 4E, large spikes in cantilever excursions can be reduced using the AML imaging technique. This can be achieved using data-driven iterative feedforward control in combination with predicted next-line sample contours and predicted next-line tracking errors in iterative control with an additional mean-offset feedback loop (for DM imaging) and set-point optimization. Improve. Consequently, the tracking of the topography of the sample can be significantly improved (see Fig. 4F), especially near the upper and lower edges. Furthermore, by using the sample topography quantification of the present disclosure (Equation (3) and Equation (3a)) utilizing tracked sample topography and cantilever offset, the sample topography obtained by AML imaging techniques can be further improved, and can be closer to the original sample morphology, as shown in Figure 4D.

Referring now to FIG. 5, an exemplary flowchart of an algorithm based on the operating principles of the present AML imaging modality in the embodiments discussed above is disclosed. In step 501, a high frequency pulsating piezoelectric may initiate probe vibration to induce probe-sample interaction. Logic may continue to 502 to use vibration demodulator 304 to measure the instantaneous amplitude of probe vibration in real time. The measurements can be provided to the vibration amplitude feedback controller 301 in step 503 so that the controller can adjust the probe vibration amplitude to the desired A _set .

In step 504 , the logic may measure the average offset in the outer loop 350 and adjust it in step 505 using the feedback controller 305 . In step 506, the logic may also track the average offset adjustment of the inner loop 360. As discussed, the optimal tuning value is determined experimentally, eg, before running the AFM system.

In step 507, the logic may also quantify the sample topography based on the z piezoelectric displacement and the mean offset. After quantization, the logic may also predict the next row of sample topography and the next row of tracking errors using the data-driven online iterative feedforward control loop 310 in step 508 and pre-drive the z-piezo based on the prediction in step 509 .

In step 510, the controller 314 can be used to optimize the vibration amplitude set point online such that the vibration set point can be updated based on the real-time measured average excursion and vibration amplitude ratio relationship in order to calculate the average excursion amplitude and vibration Uncertainty and variation in the relationship between amplitude ratios.

In step 511, the free vibration amplitude may be adjusted to minimize tip-sample interaction forces associated with vibration. The PID controller 312 can be used to update the free vibration amplitude using the difference between the real-time measured vibration amplitude and a pre-selected minimum vibration amplitude.

Referring now to FIG. 7 , an AFM system according to an embodiment includes a microcantilever 708 having a sharpened probe 711 at its end that can be used to scan the surface of a sample 712 . The cantilever is supported by a base (cantilever holder) 707 at its other end. As the probe begins to approach the sample surface, the force between the tip and the sample can cause deflection of the cantilever. The deflection of the cantilever can be measured with an optical device comprising a laser 710 and an array of photodiodes (detectors) 709 .

Scanner 701 may generate relative motion between probe 711 and sample 712 when monitoring probe-sample interactions. In certain embodiments, the scanner may be a scanning probe microscope (SPM) scanner. In this way, images or other measurements of the sample can be obtained. Scanner 701 may include one or more actuators that typically produce motion in three orthogonal directions (XYZ). The scanner 701 can be a single integrated unit, such as a piezoelectric tube actuator that moves the sample or probe in all three axes. Alternatively, the scanner may be an assembly of multiple separate actuators. In some embodiments, the scanner may be divided into multiple components, such as xy actuators that move the sample and separate z actuators that move the probe. In accordance with the disclosure discussed above, in probe scanning with respect to the sample surface, a controller 714 may be employed that includes a feedback mechanism for controlling the SPM scanner 701 to drive the z piezoelectric actuator, which may Installed in base 707 . The signal from photodiode 709 is passed to controller 714 . Feedback mechanisms include the AML imaging module discussed with respect to Figure 3A.

While the embodiments of the present disclosure employ sample tracking in the lateral x-y direction using a z piezoelectric actuator (along the vertical axis), it should be understood to those skilled in the art that, in accordance with the principles disclosed in this disclosure, Sample tracking can be performed in other directions.

The controller 614 may be part of the central processing unit of the system and may perform the calculations and logic operations required to execute the programs. A processing unit, alone or in combination with one or more other elements, may be a processing device, computing device, or processor, as such terms are used in this disclosure. As used in this document and in the claims, the term "processor" may refer to a single processor or any number of processors in a group of processors. Read only memory (ROM) and random access memory (RAM) constitute examples of memory devices. Additional memory devices may include, for example, external or internal disk drives, hard drives, flash memory, USB drives, or other types of devices used as data storage devices. As previously mentioned, the various drivers and controllers are optional devices. Additionally, the memory device may be configured to include separate files for storing any software modules or instructions, auxiliary data, event data, common files for storing sets of contingency tables and/or regression models, or for storing all of the above. One or more databases of discussed information.

Program instructions, software or interaction modules for performing any functional steps associated with the processes described above may be stored in ROM and/or RAM. Alternatively, the program instructions may be stored on a non-transitory computer readable medium such as an optical disk, digital disk, flash memory, memory card, USB drive, optical disk storage medium and/or other recording medium.

In certain embodiments, the imaging speed is selected from about 0.1 Hz to 40 Hz. In certain other embodiments, the imaging speed may depend on factors such as the size of the sample, the topography of the sample, and the type of AFM.

In an embodiment, the addition of an average offset feedback loop 310 and a feedforward controller 320 can greatly speed up the tracking of sample topography during AML imaging. Keeping the average offset around the desired value helps keep the RMS tap amplitude around the set point. Additionally, the primary average probe-sample interaction force can be minimized by keeping the tap amplitude near a level where the corresponding average shift is minimized. In addition, through the prediction and fast convergence of the optimally predicted sample topography profile, the feedforward controller further reduces the oscillation of the tap amplitude in the case of sudden sample topography changes when the scan speed is increased. Thus, the TM offset loop of the present disclosure herein tracks sample topography during imaging while keeping the average probe-sample force close to a minimum.

The above-disclosed features and functions, and alternatives, may be incorporated into many other different systems or applications. Various presently unforeseen or unanticipated alternatives, modifications, changes or improvements may be made by those skilled in the art, each of which is to be included in the embodiments of the present disclosure.

Claims (22)

1. A method of imaging a sample using a high speed dynamic mode atomic force microscope, wherein the method comprises:

scanning a tip of a cantilever probe over a surface of a sample;

adjusting the vibration amplitude of the tip via a first signal generated in a first feedback controller to remain constant at a set point value (A)_set)；

Measuring an average tap deflection of the tip for each time period;

adjusting the average tap offset by maintaining the average tap offset near a desired value and by adjusting the average vertical position of the scanning cantilever in each tap cycle via a second signal generated in a second feedback controller;

tracking and measuring an adjustment to the measured average tap offset from the desired value during adjustment; and

generating a topographical image of the sample based on the height of a point on the sample surface determined by combining the first signal, the second signal, and the measured adjustment to the mean tap offset of the cantilever probe relative to a fixed reference point.

2. The method of claim 1, wherein the high-speed dynamic mode atomic force microscope comprises one or more of: a tapping mode, a non-contact mode, and a peak-force tapping mode.

3. The method of claim 1, wherein adjusting the mean tap offset comprises: using A_setThe ratio to the free amplitude determines the required average offset; and

the measured average tap shift is adjusted to the desired average shift.

4. The method of claim 1, wherein the second feedback controller comprises an inner and outer feedback loop structure having an outer feedback loop and an inner feedback loop nested in the outer feedback loop,

wherein the outer feedback loop adjusts the average tap offset and the inner feedback loop performs tracking and measuring adjustments to the measured average tap offset during the adjusting.

5. The method of claim 4, wherein the outer feedback loop is a proportional-integral-derivative (PID) type controller having a PID parameter K_P、K_IAnd K_D。

6. The method according to claim 5, wherein the PID-type controller employs the following algorithm:

d_TM-set(j+1)＝k_Id_TM-set(j)+k_Pe_TM(j)+k_D[e_TM(j1)-e_TM(j)]

wherein e_TM(j)＝d_TM-d-d_TM(j) N-1, wherein j is 2.

7. The method of claim 6, wherein the PID parameter has the following values: k_P＝1、K_I1 and K_DWhere p is the sample point-by-point gradient factor.

8. The method of claim 7, wherein ρ < 1.

9. The method according to claim 5, wherein the PID-type controller employs the following algorithm:

d_set，0＝d_set，org，

wherein

Wherein T ∈ [ O, T_scan]，

Wherein d is_set，0Is an offset setpoint on a first scan line;

d_set，kis the offset setpoint on the k-th scan line;

d_k+1(t) is the minimum of the prediction detection at the k +1 th scan line;

D*_minis the minimum deflection/force required to maintain a stable repulsive tip-sample interaction;

T_scanis the scan period;

d_set，orgis the original offset set point selected prior to the imaging procedure; and

ρ ∈ [0, 1] is a gradient factor that can be adjusted to improve imaging quality.

10. The method of claim 3, wherein the required average offset is determined such that A_setThe ratio to the free amplitude is about 10% to 30%.

11. The method of claim 1, further comprising based on the measured average offset and a_setReal-time relationship between vibration amplitude ratio and free amplitude on-line optimization A_set。

12. The method of claim 11, further comprising predicting, via a third feedback controller, a next row sample topography and a next row tracking error for tracking the mean tap offset adjustment.

13. The method of claim 12, wherein the third feedback controller is a feed forward controller comprising a data-driven iterative learning controller.

14. The method of claim 13, wherein the feedforward controller implements the following algorithm to obtain a control input:

U_ff，0(jω)＝0，

e_k(jω)＝H_ffd，k+1(jω)-Z_k(jω)，

wherein

Is the fourier transform of the corresponding signal;

λ is a preselected constant for ensuring convergence of the iteration;

U_ff+fb，k(. h) is the overall control input applied to the z piezoelectric actuator (feedback + feedforward);

Z_k(. h) is the z-piezoelectric displacement measured for the kth scan line; and

H_ffd，k+1(. cndot.) is the predetermined trajectory that the z-piezo needs to track at the k +1 th scan line.

15. The method of claim 12, further comprising using the next row of sample topographies and the prediction of the next row tracking errors to reduce tracking errors in areas of a sample surface having features that provide abrupt dynamic changes, including cliffs and edges.

16. The method of claim 15, wherein using the next row sample topography and the prediction of the next row tracking error to reduce tracking errors comprises: the trajectory required for the next line is obtained using the following formula:

h_ffd，k+1(j)＝h_k(j)+α[d_TM，k(j)-d_T，M-d]，j＝1，...N_l·，

wherein N is₁Is the total sample point for each image row; and

α is a correction factor.

17. The method of claim 16, wherein the value of α is adjusted based on the estimated height of the feature.

18. The method of claim 17, wherein acquiring the trajectory required for the next row further comprises:

performing a repetitive scan of the first row until convergence is reached; and

using the convergence as an initial input for an iteration of a next scan line.

19. The method of claim 13, wherein the feedforward controller further comprises a zero-phase low-pass filter configured to filter noise from being fed back to the feedforward controller.

20. A method of imaging a sample using a high speed dynamic mode atomic force microscope, wherein the method comprises:

scanning a tip of a cantilever probe over a surface of a sample;

adjusting the vibration amplitude of the tip via a first signal generated in a first feedback controller to remain constant at a set point value (A)_set)；

Measuring an average tap deflection of the tip for each time period;

adjusting the average tap offset by maintaining the average tap offset near a desired value and by adjusting the average vertical position of the scanning cantilever in each tap cycle via a second signal generated in a second feedback controller;

tracking and measuring an adjustment to the measured average tap offset from the desired value during adjustment;

predicting, via a third feedback controller, a next row sample topography and a next row tracking error for tracking the mean tap offset adjustment;

using the predicted next row sample topography and next row tracking error in adjusting the average tap offset; and

generating a topographical image of the sample based on the height of a point on the sample surface determined by combining the first signal, the second signal, and the measured adjustment to the mean tap offset of the cantilever probe relative to a fixed reference point.

21. The method of claim 20, further comprising an online iterative control applied to the z piezo actuator to maintain a stable stroke.

22. The method of claim 21, wherein applying the online iterative control comprises adjusting a by online point-by-point adjustment to adjust a_set。

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